Abstract

A Chapman–Enskog closure approximation for the third order fluctuating velocity correlations in the particle phase of a turbulent dilute gas–particulate suspension is used to formulate the closed system of equations and boundary conditions for fully developed flow of dilute but densely loaded suspension of “high-inertia” particles in a vertical pipe. The particle size and concentration are assumed to be sufficiently small so that direct interparticle interactions (e.g., collisions) can be neglected, and the main mechanism inducing particle velocityfluctuations is the interaction between particles and turbulent flow. The case of moderate gas pressure gradient is studied thus modeling a number of practical applications (e.g., riser flow). The resulting set of continuum equations, consisting of mass and momentum conservation and Reynolds stress equations for the particulate phase, is free of empirical parameters. Although in a general case the effective stress in the particulate phase is anisotropic, the criterion is obtained showing that in a wide range of parameters typical for applications, this anisotropy may be neglected so that the equations for the individual diagonal components of the Reynolds stress tensor may be reduced to just one conservation equation for the particle fluctuation energy. The numerical solution shows the bifurcation of flow properties at a certain gas pressure gradient, thus providing an explicit criterion (i.e., a critical pressure gradient for the given total mass flux of solid particles) for upward particulate flow. The profiles of particle volume fraction and velocity are calculated, the former demonstrating the phenomenon of particle segregation toward the wall. A generalization of the model for the nearly developed flow is discussed, and an estimate is derived for the vertical distance required for the flow to become fully developed.